[Paper Review] From Derrida's random energy model to branching random walks: from 1 to 3

Nicola Kistler, Marius A. Schmidt|arXiv (Cornell University)|Mar 13, 2015

Stochastic processes and statistical mechanics15 references3 citations

TL;DR

This paper studies the extremal behavior of a class of Gaussian fields with hierarchical correlation structure parameterized by α ∈ [0,1], interpolating between Derrida's Random Energy Model (REM, α=0) and the Branching Random Walk (BRW, α=1). It establishes that for all α < 1, the limiting extremal process converges weakly to a Poisson point process with intensity e^{-β_c x}dx / √(2π), where β_c = √(2 log 2), indicating that the extreme values remain in the REM universality class despite increasing correlation as α increases.

ABSTRACT

We study the extremes of a class of Gaussian fields with in-built hierarchical structure. The number of scales in the underlying trees depends on a parameter alpha in [0,1]: choosing alpha=0 yields the random energy model by Derrida (REM), whereas alpha=1 corresponds to the branching random walk (BRW). When the parameter alpha increases, the level of the maximum of the field decreases smoothly from the REM- to the BRW-value. However, as long as alpha&lt;1 strictly, the limiting extremal process is always Poissonian.

Motivation & Objective

• To understand the extreme value behavior of correlated Gaussian fields with hierarchical tree structure.
• To analyze how the extremal process evolves as the correlation structure transitions from uncorrelated (REM) to highly correlated (BRW).
• To characterize the limiting distribution of the maximum and the extremal process for intermediate levels of correlation (α ∈ [0,1)).
• To clarify why the BRW case (α=1) is an exception, where the extremal process does not converge to a Poisson law.
• To establish a universal Poissonian limit for the extremal process across all α < 1, despite increasing correlation.

Proposed method

• Construct a Gaussian field X^(α,N) on a hierarchical tree with N^α scales and 2^N configurations, using independent Gaussian increments at each level with variance N^{1−α}.
• Define the energy of each configuration as the sum of Gaussian increments along its path, resulting in a field with covariances cov(X_σ, X_τ) = (σ ∧ τ) N^{1−α}, where σ ∧ τ is the overlap level.
• Use a centering sequence a_N^(α) = β_c N − (1+2α)/(2β_c) log N, with β_c = √(2 log 2), to recenter the field for studying extremal fluctuations.
• Apply weak convergence techniques to show that the recentered empirical measure Ξ_N^(α) = ∑_σ δ_{X_σ^(α,N) − a_N^(α)} converges weakly to a Poisson point process.
• Leverage properties of discrete and continuous Brownian bridges, including cyclic symmetry and monotonicity, to control the probability that a bridge stays below zero or a small ε.
• Use induction and Gaussian estimates to bound the difference in first-passage probabilities for bridges constrained below 0 versus below ε, showing decay proportional to |ε|/n.

Experimental results

Research questions

• RQ1 How does the extremal process of a hierarchical Gaussian field behave as the system size N → ∞ for α ∈ [0,1)?
• RQ2 Does the transition from REM (α=0) to BRW (α=1) result in a continuous change in the limiting extremal statistics?
• RQ3 Why does the extremal process remain Poissonian for all α < 1, despite increasing correlation?
• RQ4 What distinguishes the BRW case (α=1) from the rest of the family in terms of extreme value limits?
• RQ5 Can the probability that a Brownian bridge stays below a level ε be controlled uniformly in terms of ε and N?

Key findings

• For all α ∈ [0,1), the extremal process of the hierarchical Gaussian field converges weakly to a Poisson point process with intensity measure e^{-β_c x}dx / √(2π), where β_c = √(2 log 2).
• The centering sequence a_N^(α) = β_c N − (1+2α)/(2β_c) log N interpolates smoothly between the REM and BRW values, with the pre-factor (1+2α) accounting for the continuous decrease in maximum level as α increases.
• Despite increasing correlation with α, the limiting extremal process remains Poissonian for all α < 1, indicating that the REM universality class persists in this regime.
• The BRW case (α=1) is an exception: the limiting extremal process is not Poissonian, and the maximum does not converge to a Gumbel distribution.
• The probability that a discrete Brownian bridge of length n stays below zero is asymptotically of order 1/n, and this behavior is stable under small shifts of the barrier by ε, with the difference bounded by c|ε|/n.
• The key technical step relies on showing that the probability of a bridge violating a barrier at ε is close to that at 0, using induction and Markov property arguments with Gaussian tail estimates.